Solve 85=9^x+1

$$85 = 9 ^ { x + 1 }$$

Solve for x

$$x = \frac{\log_{3} {(85)} - 2}{2} \approx 1.021937722$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$9^{x+1}=85$$

Take the logarithm of both sides of the equation.

$$\log(9^{x+1})=\log(85)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\left(x+1\right)\log(9)=\log(85)$$

Divide both sides by \log(9).

$$x+1=\frac{\log(85)}{\log(9)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$x+1=\log_{9}\left(85\right)$$

Subtract 1 from both sides of the equation.

$$x=\frac{\log_{3}\left(85\right)}{2}-1$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(3)}+\frac{\log_{3}\left(85\right)}{2}-1$$

$$n_{1}\in \mathrm{Z}$$